Calculus Examples

$F (x) = x + 4 \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x + 4 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4 \cos (x)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [4 \cos (x)]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [4 \cos (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [4 \cos (x)]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (x)$ with respect to $x$ is $4 \frac{d}{d x} [\cos (x)]$ .

$1 + 4 \frac{d}{d x} [\cos (x)]$

Step 1.2.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$1 + 4 (- \sin (x))$

Step 1.2.3

Multiply $- 1$ by $4$ .

$1 - 4 \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $1 - 4 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 \sin (x)]$ .

$F'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- 4 \sin (x))$

Step 2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$F'' (x) = 0 + \frac{d}{d x} (- 4 \sin (x))$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 \sin (x)]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \sin (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\sin (x)]$ .

$F'' (x) = 0 - 4 \frac{d}{d x} \sin (x)$

Step 2.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$F'' (x) = 0 - 4 \cos (x)$

Step 2.3

Subtract $4 \cos (x)$ from $0$ .

$F'' (x) = - 4 \cos (x)$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$1 - 4 \sin (x) = 0$

Step 4

Subtract $1$ from both sides of the equation.

$- 4 \sin (x) = - 1$

Step 5

Divide each term in $- 4 \sin (x) = - 1$ by $- 4$ and simplify.

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Step 5.1

Divide each term in $- 4 \sin (x) = - 1$ by $- 4$ .

$\frac{- 4 \sin (x)}{- 4} = \frac{- 1}{- 4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 4 \sin (x)}{- 4} = \frac{- 1}{- 4}$

Step 5.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 1}{- 4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{1}{4}$

Step 6

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1}{4})$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arcsin (\frac{1}{4})$ .

$x = 0.25268025$

Step 8

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.25268025$

Step 9

Solve for $x$ .

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Step 9.1

Remove parentheses.

$x = 3.14159265 - 0.25268025$

Step 9.2

Remove parentheses.

$x = (3.14159265) - 0.25268025$

Step 9.3

Subtract $0.25268025$ from $3.14159265$ .

$x = 2.88891239$

Step 10

The solution to the equation $x = 0.25268025$ .

$x = 0.25268025, 2.88891239$

Step 11

Evaluate the second derivative at $x = 0.25268025$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4 \cos (0.25268025)$

Step 12

$x = 0.25268025$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0.25268025$ is a local maximum

Step 13

Find the y-value when $x = 0.25268025$ .

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Step 13.1

Replace the variable $x$ with $0.25268025$ in the expression.

$F (0.25268025) = (0.25268025) + 4 \cos (0.25268025)$

Step 13.2

Simplify the result.

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Step 13.2.1

Add $0.25268025$ and $4 \cos (0.25268025)$ .

$F (0.25268025) = 4.1256636$

Step 13.2.2

The final answer is $4.1256636$ .

$y = 4.1256636$

Step 14

Evaluate the second derivative at $x = 2.88891239$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4 \cos (2.88891239)$

Step 15

$x = 2.88891239$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2.88891239$ is a local minimum

Step 16

Find the y-value when $x = 2.88891239$ .

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Step 16.1

Replace the variable $x$ with $2.88891239$ in the expression.

$F (2.88891239) = (2.88891239) + 4 \cos (2.88891239)$

Step 16.2

Simplify the result.

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Step 16.2.1

Add $2.88891239$ and $4 \cos (2.88891239)$ .

$F (2.88891239) = - 0.98407094$

Step 16.2.2

The final answer is $- 0.98407094$ .

$y = - 0.98407094$

Step 17

These are the local extrema for $F (x) = x + 4 \cos (x)$ .

$(0.25268025, 4.1256636)$ is a local maxima

$(2.88891239, - 0.98407094)$ is a local minima

Step 18